Multiple models to capture the variability in biological neurons and networks

Abstract

How tightly tuned are the synaptic and intrinsic properties that give rise to neuron and circuit function? Experimental work shows that these properties vary considerably across identified neurons in different animals. Given this variability in experimental data, this review describes some of the complications of building computational models to aid in understanding how system dynamics arise from the interaction of system components. We argue that instead of trying to build a single model that captures the generic behavior of a neuron or circuit, it is beneficial to construct a population of models that captures the behavior of the population that provided the experimental data. Studying a population of models with different underlying structure and similar behaviors provides opportunities to discover unsuspected compensatory mechanisms that contribute to neuron and network function.

Figure 1

The pyloric rhythm has a variable period but phase relationships are held invariant. (a) Extracellular recordings from a slow pyloric rhythm showing its characteristic repeating pattern of PD, LP and PY neuron activity (on the LP and PY traces, only the largest spikes correspond to spikes from the LP and PY neurons respectively). Arrows indicate measurements made on each pyloric cycle. Gray arrow indicates pyloric period, measured as the latency from the onset of one PD neuron burst to the next. Colored arrows indicate latencies measured from the onset of the PD neuron burst. The dark blue arrow indicates the latency of PD neuron offset. The red arrow indicates the latency of LP neuron onset. The light blue arrow indicates the latency of LP neuron offset. The purple arrow indicates the latency of PY neuron onset. The pink arrow indicates the latency of PY neuron offset. These latencies were then divided by the period to give the phase relationships shown in c. (b) Extracellular recordings from a fast pyloric rhythm. Data are presented as in a. (c) Phase of burst onset/offset versus pyloric period. Each point represents one of 99 animals. Period is a mean period calculated over many cycles, as are phases. Dark blue points, phase of PD neuron offset; red points, phase of LP neuron onset; light blue points, phase of LP neuron offset; purple points, phase of PY neuron onset; pink points, phase of PY neuron offset. The histograms on top of the plot show the distribution of pyloric rhythm periods. The histograms on the right show the distributions of each of the phases, coded in color as for the data points. Adapted from ref. .

Figure 2

Example distributions of neuron parameters for neurons that all share a common behavior or set of behaviors. In all panels, dark blue dots represent individual neurons, the red cross represents the mean of the distribution and the light blue triangle represents the hypothetical neuron with all parameters set to their largest, or ‘best’, values. (a) A population with statistically independent parameters. (b) A population in which the mean is not representative. (c) A population with a strong positive correlation between parameters. (d) A population with a strong negative correlation between parameters. (e) A population with two very different subpopulations. (f) A population with a donut-shaped distribution.

Figure 3

Model LP neurons with similar behavior but substantially different parameters. (a, b) Traces from two randomly generated model LP neurons receiving ongoing pyloriclike synaptic input. (c) The parameters for the two models, which are quite different. Red and blue bars show the parameters of the model that generated the red/blue trace in a. For each parameter, a red and blue bar are superimposed, with their region of overlap shown as purple. Parameters are sorted by the absolute difference between them in the two models. ḡ parameters are maximal conductances of different currents, E parameters are reversal potentials, P̄_Ca is the maximal permeability of the Ca²⁺ current and V_{½, pr} is the half-activation voltage of a modulatory inward current. Max, maximum; min, minimum. The model is described in ref. .

Figure 4

Tolerance and degeneracy. (a) Plot of the map between maximal (max) conductance and firing rate for a hypothetical neuron with only a single variable conductance. Tolerance in the spike rate translates into tolerance in the maximal conductance, with the maximal conductance tolerance determined by the slope of the line. (b) As in a, but here the relationship between spike rate and maximal conductance has a lower slope, leading to a larger tolerance in the maximal conductance for the same spike-rate tolerance. (c) As in a and b, but with a slope of zero. In this case, the firing rate is completely insensitive to the maximal conductance, and thus the maximal conductance can take on any value. (d) Contour plot of the map between maximal conductances and spike rate for a hypothetical neuron with two variable maximal conductances. Each line denotes the set of maximal conductances that yield the given spike rate. Although each conductance affects the spike rate, there are many combinations of maximal conductances that yield the same spike rate.

Figure 5

Quantification of the effect of each model parameter on each model property for a population of LP models. The area of each circle represents the average amount of variance explained by that parameter as a fraction of the variance explained by the complete fit. The areas of the circles in each row sum to 1. Most properties show contributions from many conductances. Parameters are the same as those in Figure 3. Adapted from ref. .